Fractional Generalizations of Rodrigues-Type Formulas for Laguerre Functions in Function Spaces

Generalized Laguerre polynomials, Ln(α), verify the well-known Rodrigues’ formula. Using Weyl and Riemann–Liouville fractional calculi, we present several fractional generalizations of Rodrigues’ formula for generalized Laguerre functions and polynomials. As a consequence, we give a new addition formula and an integral representation for these polynomials. Finally, we introduce a new family of fractional Lebesgue spaces and show that some of these special functions belong to them.


Introduction
In approximation theory, the classical orthogonal polynomials of Jacobi, Laguerre, and Hermite have many properties in common, namely, the Rodrigues formula, the differential equation, the derivative formula, and the three-term recurrence relation. Under some conditions, these common properties are equivalent and characterize these classical orthogonal polynomials. See more details, for example, in [1] [Chapter 12] and [2] [Chapter V].
In [5], certain Laguerre polynomials of arbitrary orders are defined. The fractional Caputo derivative D α of order α ∈ (n − 1, n] of a function f is given there by The author defines the Laguerre polynomials Ł β α of order α > 0 by A wide generalization of Rodrigues' formula is treated in [6]. The author considers the Riemann-Liouville integral to include a large numbers of special functions, in particular Laguerre polynomials and functions [6] [Section 1].
In [7], authors use a generalization of the Rodrigues' formula to define a new special function. They study some of its properties, some recurrence relations, orthogonality property, and the continuation to the Rodrigues' formula of the Laguerre polynomials as a limit case. In addition, the confluent hypergeometric representation is given.
In this paper, we consider the Weyl and Riemann-Liouville fractional calculi, W α + and D α + with α ∈ R in the half real line in the second section. In Section 3, Theorem 1, we show the following fractional Rodrigues' formulae: where M(α, ν + 1, z) and U(α, ν + 1, z) are the confluent hypergeometric functions, This theorem extends [8] [Theorem 3] and completes the picture given in formulae [6] [(8)- (12)], where the author only considers the Riemann-Liouville fractional calculus.
In the particular case −α = n ∈ N, the confluent hypergeometric functions are essentially the Laguerre polynomials and we get a second fractional Rodrigues' formula, in Theorem 2. As a consequence, we get a new integral addition formula for Laguerre polynomials in Corollary 1. We also obtain a integral representation of W α + (t n e −λt ), i.e., and apply it to get a new integral representation of L in Theorem 3. All of these results show the deep and interesting connection between fractional calculi, in particular Weyl fractional derivation, and Laguerre polynomials.
In the last section, we introduce new fractional Lebesgue space T (α) p (t µ + t α ) which are contained in L p (R + ) with 0 ≤ µ ≤ α and p ≥ 1. Note that we understand that T (0) p (t 0 + t 0 ) = L p (R + ). As in the classical case, we show that the space T (α) p (t µ + t α ) for p > 1 is module for the algebra T 1 (t µ + t α ) (Theorem 4). This family of function spaces contains as a particular case some spaces which have appeared previously in the literature [9][10][11][12][13]. Finally, we present some special functions which belong to these fractional Lebesgue spaces T (α) p (t µ + t α ) in Remark 2.
It is easy to check that, if α = n ∈ N, then W α where f r (s) := f (rs) for r > 0; see more details in [14,15].

Proposition 1.
Take α ∈ R and f ∈ S + then : The usual convolution product * on R + is defined by for functions f , g which are "good enough", for example, absolutely integrable functions. For functions f , g ∈ S + , the following integral equality for the convolution product holds for s ∈ R + ([10,12] [Proposition 1.2]).
If α > 0 and f , g ∈ S + , we apply the Fubini theorem to get that and, if we apply W −α + (W α + g) = g, we obtain the following "integration by parts" formula: We will use both equalities in the following sections. Note that Formula (6) shows the dual behaviour between both fractional calculi.

Fractional Rodrigues' Formulae for Confluent Hypergeometric Functions
Two linearly independent solutions of Kummer's differential equation are given by confluent hypergeometric functions M(−α, 1 + ν, z) (also written by In the particular case of −α = n ∈ N, we have that There is a big amount of equalities which confluent hypergeometric functions verify. We consider the following ones: The following integral representations hold: and also the recurrence relation see all these formulae and much more in [2] [Chapter VI]. The part (i) of the next theorem includes [8] [Theorem 3] for α < 0 and ν > −1 and for α ∈ R and ν > α − 1 are presented in [6] [Section 1]. We include the proof of both parts to avoid the lack of completeness. Theorem 1. Given α ∈ R and z > 0, the following equalities hold: Proof. (i) Let ν > α − 1. For α = 0, M(0, ν + 1, z) = 1 and the equality holds. Take α > 0 and for z > 0 and by holomorphy z > 0. Finally, take α < 0. We write n = [−α] + 1 and we apply the above equality for α > 0 to get for z > 0. We apply the first formula in (7) to get and we get the equality for z > 0 and then z > 0.

Theorem 2.
Let α ∈ R, n ∈ N. Then, Proof. We give a proof by induction. Take α > 0; for n = 1, we apply Proposition 1 to get Taking the case n + 1 and again by Proposition 1, we obtain and, using induction hypothesis, We apply the recurrence Formula (2) and and we get the equality. In the case α < 0, we work in a similar way.

Remark 1.
In the case α = m ∈ N, we obtain an equivalent formula to To check this, we use L If λ > 0, we also get W α + (t n e −λt )(x) = λ α−n e −λx (−1) n n!L The addition formula for Laguerre polynomials states that ( [2] [p. 249]). The following corollary shows a new integral addition formula for Laguerre polynomials.

Corollary 1.
Let α ∈ R and n, m ∈ N. Then, Proof. We write p n (s) = s n n! e −s for s ∈ R + . Note that p n * p m = p n+m+1 for n, m ∈ N. By Theorem 2, W α (p n )(s) = (−1) n e −s L (α−n) n (s) for α, s ∈ R and n ∈ N. Finally, we apply Formula (5) to conclude the equality. Now, we want to give another representation, an integral representation of W α + (t n e −λt ) and L (α−n) n . To do this, we check the following Lemma about the Pochhammer symbol (α) j , where (α) j := α(α + 1) . .
Lemma 1. Let l ∈ N and α ∈ C. Then, it is enough to prove that P l is a polynomial in α of degree l, the leading coefficient (−1) l /l! and roots {0, 1, . . . , l − 1} where Let j ∈ {1, . . . , l − 1} and we consider the polynomial: We derive j times and evaluate in x = 1 to obtain l ∑ k=0 l k (−1) k (k + j − 1) . . . k = 0, and then and we conclude the proof.
Theorem 3. If n ∈ N and λ, α > 0, we have that Proof. By Lemma 1, we have that for l ≥ 1 and 0 ≤ j ≤ n. By the second remark in Theorem 2 and taking in the last expression l = n − j, we have that In the other hand, we apply Newton's formula, to get the equality: with t ≥ 0. From here, we have that W α + (r n e −λr )(t) = λ α e −λt t n + λ α n k ∞ t (r − t) k−1 r n−k e −λr dr, and we obtain the equality. As a consequence, we apply Theorem 2 and the first equality to obtain the second one.
As a corollary of this theorem, we show an equality considered in [14] [p. 315]. However, first, we need to comment some aspects about Laplace transform. Given f ∈ S + , the Laplace transform of f , L( f ), is given by see, for example [17]. If f is a function in two variables f = f (t, s), L( f ; t) and L( f ; s) are Laplace transforms, if there exist, in each parameter. If we apply equality (6) in the integral representation of the Laplace transform, we get that, for with f ∈ S + and where In fact, function S α (t, z) may be defined for any z ∈ C and Corollary 2. Let a, α > 0, λ > a and n ∈ N ∪ {0}. Then, we have that Proof. By the equality (6) and Theorem 3, we get that We apply the Fubini theorem to get n k ∞ 0 r n−k e −λr S α+k (r, −a)dr, and we conclude the equality.
In the next proposition, we present some results for spaces T